which implies that the operator \(\Gamma \)  is a linear self-adjoint operator.
Note that in this setting, Mercer's theorem is still valid (see [1]}), i.e., there exists a sequence of orthonormal functions \((\textbf {f}_\ell )_{\ell \in \mathbb {N}^*}\)  and positive numbers \((\mu _\ell )_{\ell \in \mathbb {N}^*}\)  (eigenfunctions and the associated eigenvalues) such that
\(\Gamma &=\sum _{\ell \in \mathbb {N}^*}\mu _\ell \textbf {f}_\ell \otimes \textbf {f}_\ell \\&=\sum _{\ell \in \mathbb {N}^*} \textbf {g}_\ell \otimes \textbf {g}_\ell ,\) 
